Bessel Functions - 10.65 Power Series

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
10.65#Ex1 ber ν ⁡ x = ( 1 2 ⁢ x ) ν ⁢ ∑ k = 0 ∞ cos ⁡ ( 3 4 ⁢ ν ⁢ π + 1 2 ⁢ k ⁢ π ) k ! ⁢ Γ ⁡ ( ν + k + 1 ) ⁢ ( 1 4 ⁢ x 2 ) k Kelvin-ber 𝜈 𝑥 superscript 1 2 𝑥 𝜈 superscript subscript 𝑘 0 3 4 𝜈 𝜋 1 2 𝑘 𝜋 𝑘 Euler-Gamma 𝜈 𝑘 1 superscript 1 4 superscript 𝑥 2 𝑘 {\displaystyle{\displaystyle\operatorname{ber}_{\nu}x=(\tfrac{1}{2}x)^{\nu}% \sum_{k=0}^{\infty}\frac{\cos\left(\frac{3}{4}\nu\pi+\frac{1}{2}k\pi\right)}{k% !\Gamma\left(\nu+k+1\right)}(\tfrac{1}{4}x^{2})^{k}}}
\Kelvinber{\nu}@@{x} = (\tfrac{1}{2}x)^{\nu}\sum_{k=0}^{\infty}\frac{\cos@{\frac{3}{4}\nu\pi+\frac{1}{2}k\pi}}{k!\EulerGamma@{\nu+k+1}}(\tfrac{1}{4}x^{2})^{k}		 ℜ ⁡ ( ν + k + 1 ) > 0 𝜈 𝑘 1 0 {\displaystyle{\displaystyle\Re(\nu+k+1)>0}} 
KelvinBer(nu, x) = ((1)/(2)*x)^(nu)* sum((cos((3)/(4)*nu*Pi +(1)/(2)*k*Pi))/(factorial(k)*GAMMA(nu + k + 1))*((1)/(4)*(x)^(2))^(k), k = 0..infinity)

KelvinBer[\[Nu], x] == (Divide[1,2]*x)^\[Nu]* Sum[Divide[Cos[Divide[3,4]*\[Nu]*Pi +Divide[1,2]*k*Pi],(k)!*Gamma[\[Nu]+ k + 1]]*(Divide[1,4]*(x)^(2))^(k), {k, 0, Infinity}, GenerateConditions->None]
Failure Failure Successful [Tested: 30] Successful [Tested: 30]
10.65#Ex2 bei ν ⁡ x = ( 1 2 ⁢ x ) ν ⁢ ∑ k = 0 ∞ sin ⁡ ( 3 4 ⁢ ν ⁢ π + 1 2 ⁢ k ⁢ π ) k ! ⁢ Γ ⁡ ( ν + k + 1 ) ⁢ ( 1 4 ⁢ x 2 ) k Kelvin-bei 𝜈 𝑥 superscript 1 2 𝑥 𝜈 superscript subscript 𝑘 0 3 4 𝜈 𝜋 1 2 𝑘 𝜋 𝑘 Euler-Gamma 𝜈 𝑘 1 superscript 1 4 superscript 𝑥 2 𝑘 {\displaystyle{\displaystyle\operatorname{bei}_{\nu}x=(\tfrac{1}{2}x)^{\nu}% \sum_{k=0}^{\infty}\frac{\sin\left(\frac{3}{4}\nu\pi+\frac{1}{2}k\pi\right)}{k% !\Gamma\left(\nu+k+1\right)}(\tfrac{1}{4}x^{2})^{k}}}
\Kelvinbei{\nu}@@{x} = (\tfrac{1}{2}x)^{\nu}\sum_{k=0}^{\infty}\frac{\sin@{\frac{3}{4}\nu\pi+\frac{1}{2}k\pi}}{k!\EulerGamma@{\nu+k+1}}(\tfrac{1}{4}x^{2})^{k}		 ℜ ⁡ ( ν + k + 1 ) > 0 𝜈 𝑘 1 0 {\displaystyle{\displaystyle\Re(\nu+k+1)>0}} 
KelvinBei(nu, x) = ((1)/(2)*x)^(nu)* sum((sin((3)/(4)*nu*Pi +(1)/(2)*k*Pi))/(factorial(k)*GAMMA(nu + k + 1))*((1)/(4)*(x)^(2))^(k), k = 0..infinity)

KelvinBei[\[Nu], x] == (Divide[1,2]*x)^\[Nu]* Sum[Divide[Sin[Divide[3,4]*\[Nu]*Pi +Divide[1,2]*k*Pi],(k)!*Gamma[\[Nu]+ k + 1]]*(Divide[1,4]*(x)^(2))^(k), {k, 0, Infinity}, GenerateConditions->None]
Failure Failure Successful [Tested: 30] Successful [Tested: 30]
10.65#Ex3 ber ⁡ x = 1 - ( 1 4 ⁢ x 2 ) 2 ( 2 ! ) 2 + ( 1 4 ⁢ x 2 ) 4 ( 4 ! ) 2 - ⋯ Kelvin-ber 𝑥 1 superscript 1 4 superscript 𝑥 2 2 superscript 2 2 superscript 1 4 superscript 𝑥 2 4 superscript 4 2 ⋯ {\displaystyle{\displaystyle\operatorname{ber}x=1-\frac{(\frac{1}{4}x^{2})^{2}% }{(2!)^{2}}+\frac{(\frac{1}{4}x^{2})^{4}}{(4!)^{2}}-\cdots}}
\Kelvinber{}@@{x} = 1-\frac{(\frac{1}{4}x^{2})^{2}}{(2!)^{2}}+\frac{(\frac{1}{4}x^{2})^{4}}{(4!)^{2}}-\dotsb		 

KelvinBer(, x) = 1 -(((1)/(4)*(x)^(2))^(2))/((factorial(2))^(2))+(((1)/(4)*(x)^(2))^(4))/((factorial(4))^(2))- ..

KelvinBer[, x] == 1 -Divide[(Divide[1,4]*(x)^(2))^(2),((2)!)^(2)]+Divide[(Divide[1,4]*(x)^(2))^(4),((4)!)^(2)]- \[Ellipsis]
Error Failure -
Failed [3 / 3]
Result: Plus[-0.921072244644165, …, KelvinBer[Null, 1.5]]
Test Values: {Rule[x, 1.5]}


Result: Plus[-0.9990234639909532, …, KelvinBer[Null, 0.5]]
Test Values: {Rule[x, 0.5]}

... skip entries to safe data
10.65#Ex4 bei ⁡ x = 1 4 ⁢ x 2 - ( 1 4 ⁢ x 2 ) 3 ( 3 ! ) 2 + ( 1 4 ⁢ x 2 ) 5 ( 5 ! ) 2 - ⋯ Kelvin-bei 𝑥 1 4 superscript 𝑥 2 superscript 1 4 superscript 𝑥 2 3 superscript 3 2 superscript 1 4 superscript 𝑥 2 5 superscript 5 2 ⋯ {\displaystyle{\displaystyle\operatorname{bei}x=\tfrac{1}{4}x^{2}-\frac{(\frac% {1}{4}x^{2})^{3}}{(3!)^{2}}+\frac{(\frac{1}{4}x^{2})^{5}}{(5!)^{2}}-\cdots}}
\Kelvinbei{}@@{x} = \tfrac{1}{4}x^{2}-\frac{(\frac{1}{4}x^{2})^{3}}{(3!)^{2}}+\frac{(\frac{1}{4}x^{2})^{5}}{(5!)^{2}}-\dotsi		 

KelvinBei(, x) = (1)/(4)*(x)^(2)-(((1)/(4)*(x)^(2))^(3))/((factorial(3))^(2))+(((1)/(4)*(x)^(2))^(5))/((factorial(5))^(2))- ..

KelvinBei[, x] == Divide[1,4]*(x)^(2)-Divide[(Divide[1,4]*(x)^(2))^(3),((3)!)^(2)]+Divide[(Divide[1,4]*(x)^(2))^(5),((5)!)^(2)]- \[Ellipsis]
Error Failure -
Failed [3 / 3]
Result: Plus[-0.5575600630044937, …, KelvinBei[Null, 1.5]]
Test Values: {Rule[x, 1.5]}


Result: Plus[-0.06249321838219961, …, KelvinBei[Null, 0.5]]
Test Values: {Rule[x, 0.5]}

... skip entries to safe data
10.65.E3 ker n ⁡ x = 1 2 ⁢ ( 1 2 ⁢ x ) - n ⁢ ∑ k = 0 n - 1 ( n - k - 1 ) ! k ! ⁢ cos ⁡ ( 3 4 ⁢ n ⁢ π + 1 2 ⁢ k ⁢ π ) ⁢ ( 1 4 ⁢ x 2 ) k - ln ⁡ ( 1 2 ⁢ x ) ⁢ ber n ⁡ x + 1 4 ⁢ π ⁢ bei n ⁡ x + 1 2 ⁢ ( 1 2 ⁢ x ) n ⁢ ∑ k = 0 ∞ ψ ⁡ ( k + 1 ) + ψ ⁡ ( n + k + 1 ) k ! ⁢ ( n + k ) ! ⁢ cos ⁡ ( 3 4 ⁢ n ⁢ π + 1 2 ⁢ k ⁢ π ) ⁢ ( 1 4 ⁢ x 2 ) k Kelvin-ker 𝑛 𝑥 1 2 superscript 1 2 𝑥 𝑛 superscript subscript 𝑘 0 𝑛 1 𝑛 𝑘 1 𝑘 3 4 𝑛 𝜋 1 2 𝑘 𝜋 superscript 1 4 superscript 𝑥 2 𝑘 1 2 𝑥 Kelvin-ber 𝑛 𝑥 1 4 𝜋 Kelvin-bei 𝑛 𝑥 1 2 superscript 1 2 𝑥 𝑛 superscript subscript 𝑘 0 digamma 𝑘 1 digamma 𝑛 𝑘 1 𝑘 𝑛 𝑘 3 4 𝑛 𝜋 1 2 𝑘 𝜋 superscript 1 4 superscript 𝑥 2 𝑘 {\displaystyle{\displaystyle\operatorname{ker}_{n}x=\tfrac{1}{2}(\tfrac{1}{2}x% )^{-n}\sum_{k=0}^{n-1}\frac{(n-k-1)!}{k!}\cos\left(\tfrac{3}{4}n\pi+\tfrac{1}{% 2}k\pi\right)(\tfrac{1}{4}x^{2})^{k}-\ln\left(\tfrac{1}{2}x\right)% \operatorname{ber}_{n}x+\tfrac{1}{4}\pi\operatorname{bei}_{n}x+\tfrac{1}{2}(% \tfrac{1}{2}x)^{n}\sum_{k=0}^{\infty}\frac{\psi\left(k+1\right)+\psi\left(n+k+% 1\right)}{k!(n+k)!}\cos\left(\tfrac{3}{4}n\pi+\tfrac{1}{2}k\pi\right)(\tfrac{1% }{4}x^{2})^{k}}}
\Kelvinker{n}@@{x} = \tfrac{1}{2}(\tfrac{1}{2}x)^{-n}\sum_{k=0}^{n-1}\frac{(n-k-1)!}{k!}\cos@{\tfrac{3}{4}n\pi+\tfrac{1}{2}k\pi}(\tfrac{1}{4}x^{2})^{k}-\ln@{\tfrac{1}{2}x}\Kelvinber{n}@@{x}+\tfrac{1}{4}\pi\Kelvinbei{n}@@{x}+\tfrac{1}{2}(\tfrac{1}{2}x)^{n}\sum_{k=0}^{\infty}\frac{\digamma@{k+1}+\digamma@{n+k+1}}{k!(n+k)!}\cos@{\tfrac{3}{4}n\pi+\tfrac{1}{2}k\pi}(\tfrac{1}{4}x^{2})^{k}		 ℜ ⁡ ( n + k + 1 ) > 0 𝑛 𝑘 1 0 {\displaystyle{\displaystyle\Re(n+k+1)>0}} 
KelvinKer(n, x) = (1)/(2)*((1)/(2)*x)^(- n)* sum((factorial(n - k - 1))/(factorial(k))*cos((3)/(4)*n*Pi +(1)/(2)*k*Pi)*((1)/(4)*(x)^(2))^(k), k = 0..n - 1)- ln((1)/(2)*x)*KelvinBer(n, x)+(1)/(4)*Pi*KelvinBei(n, x)+(1)/(2)*((1)/(2)*x)^(n)* sum((Psi(k + 1)+ Psi(n + k + 1))/(factorial(k)*factorial(n + k))*cos((3)/(4)*n*Pi +(1)/(2)*k*Pi)*((1)/(4)*(x)^(2))^(k), k = 0..infinity)

KelvinKer[n, x] == Divide[1,2]*(Divide[1,2]*x)^(- n)* Sum[Divide[(n - k - 1)!,(k)!]*Cos[Divide[3,4]*n*Pi +Divide[1,2]*k*Pi]*(Divide[1,4]*(x)^(2))^(k), {k, 0, n - 1}, GenerateConditions->None]- Log[Divide[1,2]*x]*KelvinBer[n, x]+Divide[1,4]*Pi*KelvinBei[n, x]+Divide[1,2]*(Divide[1,2]*x)^(n)* Sum[Divide[PolyGamma[k + 1]+ PolyGamma[n + k + 1],(k)!*(n + k)!]*Cos[Divide[3,4]*n*Pi +Divide[1,2]*k*Pi]*(Divide[1,4]*(x)^(2))^(k), {k, 0, Infinity}, GenerateConditions->None]
Aborted Aborted Skipped - Because timed out
Failed [9 / 9]
Result: Indeterminate
Test Values: {Rule[n, 1], Rule[x, 1.5]}


Result: Indeterminate
Test Values: {Rule[n, 2], Rule[x, 1.5]}

... skip entries to safe data
10.65.E4 kei n ⁡ x = - 1 2 ⁢ ( 1 2 ⁢ x ) - n ⁢ ∑ k = 0 n - 1 ( n - k - 1 ) ! k ! ⁢ sin ⁡ ( 3 4 ⁢ n ⁢ π + 1 2 ⁢ k ⁢ π ) ⁢ ( 1 4 ⁢ x 2 ) k - ln ⁡ ( 1 2 ⁢ x ) ⁢ bei n ⁡ x - 1 4 ⁢ π ⁢ ber n ⁡ x + 1 2 ⁢ ( 1 2 ⁢ x ) n ⁢ ∑ k = 0 ∞ ψ ⁡ ( k + 1 ) + ψ ⁡ ( n + k + 1 ) k ! ⁢ ( n + k ) ! ⁢ sin ⁡ ( 3 4 ⁢ n ⁢ π + 1 2 ⁢ k ⁢ π ) ⁢ ( 1 4 ⁢ x 2 ) k Kelvin-kei 𝑛 𝑥 1 2 superscript 1 2 𝑥 𝑛 superscript subscript 𝑘 0 𝑛 1 𝑛 𝑘 1 𝑘 3 4 𝑛 𝜋 1 2 𝑘 𝜋 superscript 1 4 superscript 𝑥 2 𝑘 1 2 𝑥 Kelvin-bei 𝑛 𝑥 1 4 𝜋 Kelvin-ber 𝑛 𝑥 1 2 superscript 1 2 𝑥 𝑛 superscript subscript 𝑘 0 digamma 𝑘 1 digamma 𝑛 𝑘 1 𝑘 𝑛 𝑘 3 4 𝑛 𝜋 1 2 𝑘 𝜋 superscript 1 4 superscript 𝑥 2 𝑘 {\displaystyle{\displaystyle\operatorname{kei}_{n}x=-\tfrac{1}{2}(\tfrac{1}{2}% x)^{-n}\sum_{k=0}^{n-1}\frac{(n-k-1)!}{k!}\sin\left(\tfrac{3}{4}n\pi+\tfrac{1}% {2}k\pi\right)(\tfrac{1}{4}x^{2})^{k}-\ln\left(\tfrac{1}{2}x\right)% \operatorname{bei}_{n}x-\tfrac{1}{4}\pi\operatorname{ber}_{n}x+\tfrac{1}{2}(% \tfrac{1}{2}x)^{n}\sum_{k=0}^{\infty}\frac{\psi\left(k+1\right)+\psi\left(n+k+% 1\right)}{k!(n+k)!}\sin\left(\tfrac{3}{4}n\pi+\tfrac{1}{2}k\pi\right)(\tfrac{1% }{4}x^{2})^{k}}}
\Kelvinkei{n}@@{x} = -\tfrac{1}{2}(\tfrac{1}{2}x)^{-n}\sum_{k=0}^{n-1}\frac{(n-k-1)!}{k!}\sin@{\tfrac{3}{4}n\pi+\tfrac{1}{2}k\pi}(\tfrac{1}{4}x^{2})^{k}-\ln@{\tfrac{1}{2}x}\Kelvinbei{n}@@{x}-\tfrac{1}{4}\pi\Kelvinber{n}@@{x}+\tfrac{1}{2}(\tfrac{1}{2}x)^{n}\sum_{k=0}^{\infty}\frac{\digamma@{k+1}+\digamma@{n+k+1}}{k!(n+k)!}\sin@{\tfrac{3}{4}n\pi+\tfrac{1}{2}k\pi}(\tfrac{1}{4}x^{2})^{k}		 ℜ ⁡ ( n + k + 1 ) > 0 𝑛 𝑘 1 0 {\displaystyle{\displaystyle\Re(n+k+1)>0}} 
KelvinKei(n, x) = -(1)/(2)*((1)/(2)*x)^(- n)* sum((factorial(n - k - 1))/(factorial(k))*sin((3)/(4)*n*Pi +(1)/(2)*k*Pi)*((1)/(4)*(x)^(2))^(k), k = 0..n - 1)- ln((1)/(2)*x)*KelvinBei(n, x)-(1)/(4)*Pi*KelvinBer(n, x)+(1)/(2)*((1)/(2)*x)^(n)* sum((Psi(k + 1)+ Psi(n + k + 1))/(factorial(k)*factorial(n + k))*sin((3)/(4)*n*Pi +(1)/(2)*k*Pi)*((1)/(4)*(x)^(2))^(k), k = 0..infinity)

KelvinKei[n, x] == -Divide[1,2]*(Divide[1,2]*x)^(- n)* Sum[Divide[(n - k - 1)!,(k)!]*Sin[Divide[3,4]*n*Pi +Divide[1,2]*k*Pi]*(Divide[1,4]*(x)^(2))^(k), {k, 0, n - 1}, GenerateConditions->None]- Log[Divide[1,2]*x]*KelvinBei[n, x]-Divide[1,4]*Pi*KelvinBer[n, x]+Divide[1,2]*(Divide[1,2]*x)^(n)* Sum[Divide[PolyGamma[k + 1]+ PolyGamma[n + k + 1],(k)!*(n + k)!]*Sin[Divide[3,4]*n*Pi +Divide[1,2]*k*Pi]*(Divide[1,4]*(x)^(2))^(k), {k, 0, Infinity}, GenerateConditions->None]
Aborted Aborted Skipped - Because timed out
Failed [9 / 9]
Result: Indeterminate
Test Values: {Rule[n, 1], Rule[x, 1.5]}


Result: Indeterminate
Test Values: {Rule[n, 2], Rule[x, 1.5]}

... skip entries to safe data
10.65#Ex5 ker ⁡ x = - ln ⁡ ( 1 2 ⁢ x ) ⁢ ber ⁡ x + 1 4 ⁢ π ⁢ bei ⁡ x + ∑ k = 0 ∞ ( - 1 ) k ⁢ ψ ⁡ ( 2 ⁢ k + 1 ) ( ( 2 ⁢ k ) ! ) 2 ⁢ ( 1 4 ⁢ x 2 ) 2 ⁢ k Kelvin-ker 𝑥 1 2 𝑥 Kelvin-ber 𝑥 1 4 𝜋 Kelvin-bei 𝑥 superscript subscript 𝑘 0 superscript 1 𝑘 digamma 2 𝑘 1 superscript 2 𝑘 2 superscript 1 4 superscript 𝑥 2 2 𝑘 {\displaystyle{\displaystyle\operatorname{ker}x=-\ln\left(\tfrac{1}{2}x\right)% \operatorname{ber}x+\tfrac{1}{4}\pi\operatorname{bei}x+\sum_{k=0}^{\infty}(-1)% ^{k}\frac{\psi\left(2k+1\right)}{((2k)!)^{2}}(\tfrac{1}{4}x^{2})^{2k}}}
\Kelvinker{}@@{x} = -\ln@{\tfrac{1}{2}x}\Kelvinber{}@@{x}+\tfrac{1}{4}\pi\Kelvinbei{}@@{x}+\sum_{k=0}^{\infty}(-1)^{k}\frac{\digamma@{2k+1}}{((2k)!)^{2}}(\tfrac{1}{4}x^{2})^{2k}		 

KelvinKer(, x) = - ln((1)/(2)*x)*KelvinBer(, x)+(1)/(4)*Pi*KelvinBei(, x)+ sum((- 1)^(k)*(Psi(2*k + 1))/((factorial(2*k))^(2))*((1)/(4)*(x)^(2))^(2*k), k = 0..infinity)

KelvinKer[, x] == - Log[Divide[1,2]*x]*KelvinBer[, x]+Divide[1,4]*Pi*KelvinBei[, x]+ Sum[(- 1)^(k)*Divide[PolyGamma[2*k + 1],((2*k)!)^(2)]*(Divide[1,4]*(x)^(2))^(2*k), {k, 0, Infinity}, GenerateConditions->None]
Error Aborted - Skipped - Because timed out
10.65#Ex6 kei ⁡ x = - ln ⁡ ( 1 2 ⁢ x ) ⁢ bei ⁡ x - 1 4 ⁢ π ⁢ ber ⁡ x + ∑ k = 0 ∞ ( - 1 ) k ⁢ ψ ⁡ ( 2 ⁢ k + 2 ) ( ( 2 ⁢ k + 1 ) ! ) 2 ⁢ ( 1 4 ⁢ x 2 ) 2 ⁢ k + 1 Kelvin-kei 𝑥 1 2 𝑥 Kelvin-bei 𝑥 1 4 𝜋 Kelvin-ber 𝑥 superscript subscript 𝑘 0 superscript 1 𝑘 digamma 2 𝑘 2 superscript 2 𝑘 1 2 superscript 1 4 superscript 𝑥 2 2 𝑘 1 {\displaystyle{\displaystyle\operatorname{kei}x=-\ln\left(\tfrac{1}{2}x\right)% \operatorname{bei}x-\tfrac{1}{4}\pi\operatorname{ber}x+\sum_{k=0}^{\infty}(-1)% ^{k}\frac{\psi\left(2k+2\right)}{((2k+1)!)^{2}}(\tfrac{1}{4}x^{2})^{2k+1}}}
\Kelvinkei{}@@{x} = -\ln@{\tfrac{1}{2}x}\Kelvinbei{}@@{x}-\tfrac{1}{4}\pi\Kelvinber{}@@{x}+\sum_{k=0}^{\infty}(-1)^{k}\frac{\digamma@{2k+2}}{((2k+1)!)^{2}}(\tfrac{1}{4}x^{2})^{2k+1}		 

KelvinKei(, x) = - ln((1)/(2)*x)*KelvinBei(, x)-(1)/(4)*Pi*KelvinBer(, x)+ sum((- 1)^(k)*(Psi(2*k + 2))/((factorial(2*k + 1))^(2))*((1)/(4)*(x)^(2))^(2*k + 1), k = 0..infinity)

KelvinKei[, x] == - Log[Divide[1,2]*x]*KelvinBei[, x]-Divide[1,4]*Pi*KelvinBer[, x]+ Sum[(- 1)^(k)*Divide[PolyGamma[2*k + 2],((2*k + 1)!)^(2)]*(Divide[1,4]*(x)^(2))^(2*k + 1), {k, 0, Infinity}, GenerateConditions->None]
Error Failure -
Failed [3 / 3]
Result: Plus[-0.23161280473545226, Times[-1.0, KelvinBer[Null, 1.5]], KelvinKei[Null, 1.5]]
Test Values: {Rule[x, 1.5]}


Result: Plus[-0.02641550246351669, Times[-1.0, KelvinBer[Null, 0.5]], KelvinKei[Null, 0.5]]
Test Values: {Rule[x, 0.5]}

... skip entries to safe data
10.65.E6 ber ν 2 ⁡ x + bei ν 2 ⁡ x = ( 1 2 ⁢ x ) 2 ⁢ ν ⁢ ∑ k = 0 ∞ 1 Γ ⁡ ( ν + k + 1 ) ⁢ Γ ⁡ ( ν + 2 ⁢ k + 1 ) ⁢ ( 1 4 ⁢ x 2 ) 2 ⁢ k k ! Kelvin-ber 𝜈 2 𝑥 Kelvin-bei 𝜈 2 𝑥 superscript 1 2 𝑥 2 𝜈 superscript subscript 𝑘 0 1 Euler-Gamma 𝜈 𝑘 1 Euler-Gamma 𝜈 2 𝑘 1 superscript 1 4 superscript 𝑥 2 2 𝑘 𝑘 {\displaystyle{\displaystyle{\operatorname{ber}_{\nu}^{2}}x+{\operatorname{bei% }_{\nu}^{2}}x=(\tfrac{1}{2}x)^{2\nu}\sum_{k=0}^{\infty}\frac{1}{\Gamma\left(% \nu+k+1\right)\Gamma\left(\nu+2k+1\right)}\frac{(\frac{1}{4}x^{2})^{2k}}{k!}}}
\Kelvinber{\nu}^{2}@@{x}+\Kelvinbei{\nu}^{2}@@{x} = (\tfrac{1}{2}x)^{2\nu}\sum_{k=0}^{\infty}\frac{1}{\EulerGamma@{\nu+k+1}\EulerGamma@{\nu+2k+1}}\frac{(\frac{1}{4}x^{2})^{2k}}{k!}		 ℜ ⁡ ( ν + k + 1 ) > 0 , ℜ ⁡ ( ν + 2 ⁢ k + 1 ) > 0 formulae-sequence 𝜈 𝑘 1 0 𝜈 2 𝑘 1 0 {\displaystyle{\displaystyle\Re(\nu+k+1)>0,\Re(\nu+2k+1)>0}} 
(KelvinBer(nu, x))^(2)+ (KelvinBei(nu, x))^(2) = ((1)/(2)*x)^(2*nu)* sum((1)/(GAMMA(nu + k + 1)*GAMMA(nu + 2*k + 1))*(((1)/(4)*(x)^(2))^(2*k))/(factorial(k)), k = 0..infinity)

(KelvinBer[\[Nu], x])^(2)+ (KelvinBei[\[Nu], x])^(2) == (Divide[1,2]*x)^(2*\[Nu])* Sum[Divide[1,Gamma[\[Nu]+ k + 1]*Gamma[\[Nu]+ 2*k + 1]]*Divide[(Divide[1,4]*(x)^(2))^(2*k),(k)!], {k, 0, Infinity}, GenerateConditions->None]
Successful Successful - Successful [Tested: 30]
10.65.E7 ber ν ⁡ x ⁢ bei ν ′ ⁡ x - ber ν ′ ⁡ x ⁢ bei ν ⁡ x = ( 1 2 ⁢ x ) 2 ⁢ ν + 1 ⁢ ∑ k = 0 ∞ 1 Γ ⁡ ( ν + k + 1 ) ⁢ Γ ⁡ ( ν + 2 ⁢ k + 2 ) ⁢ ( 1 4 ⁢ x 2 ) 2 ⁢ k k ! Kelvin-ber 𝜈 𝑥 diffop Kelvin-bei 𝜈 1 𝑥 diffop Kelvin-ber 𝜈 1 𝑥 Kelvin-bei 𝜈 𝑥 superscript 1 2 𝑥 2 𝜈 1 superscript subscript 𝑘 0 1 Euler-Gamma 𝜈 𝑘 1 Euler-Gamma 𝜈 2 𝑘 2 superscript 1 4 superscript 𝑥 2 2 𝑘 𝑘 {\displaystyle{\displaystyle\operatorname{ber}_{\nu}x\operatorname{bei}_{\nu}'% x-\operatorname{ber}_{\nu}'x\operatorname{bei}_{\nu}x=(\tfrac{1}{2}x)^{2\nu+1}% \sum_{k=0}^{\infty}\frac{1}{\Gamma\left(\nu+k+1\right)\Gamma\left(\nu+2k+2% \right)}\frac{(\frac{1}{4}x^{2})^{2k}}{k!}}}
\Kelvinber{\nu}@@{x}\Kelvinbei{\nu}'@@{x}-\Kelvinber{\nu}'@@{x}\Kelvinbei{\nu}@@{x} = (\tfrac{1}{2}x)^{2\nu+1}\sum_{k=0}^{\infty}\frac{1}{\EulerGamma@{\nu+k+1}\EulerGamma@{\nu+2k+2}}\frac{(\frac{1}{4}x^{2})^{2k}}{k!}		 ℜ ⁡ ( ν + k + 1 ) > 0 , ℜ ⁡ ( ν + 2 ⁢ k + 2 ) > 0 formulae-sequence 𝜈 𝑘 1 0 𝜈 2 𝑘 2 0 {\displaystyle{\displaystyle\Re(\nu+k+1)>0,\Re(\nu+2k+2)>0}} 
KelvinBer(nu, x)*diff( KelvinBei(nu, x), x$(1) )- diff( KelvinBer(nu, x), x$(1) )*KelvinBei(nu, x) = ((1)/(2)*x)^(2*nu + 1)* sum((1)/(GAMMA(nu + k + 1)*GAMMA(nu + 2*k + 2))*(((1)/(4)*(x)^(2))^(2*k))/(factorial(k)), k = 0..infinity)

KelvinBer[\[Nu], x]*D[KelvinBei[\[Nu], x], {x, 1}]- D[KelvinBer[\[Nu], x], {x, 1}]*KelvinBei[\[Nu], x] == (Divide[1,2]*x)^(2*\[Nu]+ 1)* Sum[Divide[1,Gamma[\[Nu]+ k + 1]*Gamma[\[Nu]+ 2*k + 2]]*Divide[(Divide[1,4]*(x)^(2))^(2*k),(k)!], {k, 0, Infinity}, GenerateConditions->None]
Failure Successful
Failed [21 / 30]
Result: .7271930e-3+.45983036e-2*I
Test Values: {nu = 1/2*3^(1/2)+1/2*I, x = 3/2}


Result: -.41528503e-2+.322695404e-1*I
Test Values: {nu = 1/2*3^(1/2)+1/2*I, x = 2}

... skip entries to safe data
Failed [3 / 30]
Result: Indeterminate
Test Values: {Rule[x, 1.5], Rule[ν, -2]}


Result: Indeterminate
Test Values: {Rule[x, 0.5], Rule[ν, -2]}

... skip entries to safe data
10.65.E8 ber ν ⁡ x ⁢ ber ν ′ ⁡ x + bei ν ⁡ x ⁢ bei ν ′ ⁡ x = 1 2 ⁢ ( 1 2 ⁢ x ) 2 ⁢ ν - 1 ⁢ ∑ k = 0 ∞ 1 Γ ⁡ ( ν + k + 1 ) ⁢ Γ ⁡ ( ν + 2 ⁢ k ) ⁢ ( 1 4 ⁢ x 2 ) 2 ⁢ k k ! Kelvin-ber 𝜈 𝑥 diffop Kelvin-ber 𝜈 1 𝑥 Kelvin-bei 𝜈 𝑥 diffop Kelvin-bei 𝜈 1 𝑥 1 2 superscript 1 2 𝑥 2 𝜈 1 superscript subscript 𝑘 0 1 Euler-Gamma 𝜈 𝑘 1 Euler-Gamma 𝜈 2 𝑘 superscript 1 4 superscript 𝑥 2 2 𝑘 𝑘 {\displaystyle{\displaystyle\operatorname{ber}_{\nu}x\operatorname{ber}_{\nu}'% x+\operatorname{bei}_{\nu}x\operatorname{bei}_{\nu}'x=\tfrac{1}{2}(\tfrac{1}{2% }x)^{2\nu-1}\sum_{k=0}^{\infty}\frac{1}{\Gamma\left(\nu+k+1\right)\Gamma\left(% \nu+2k\right)}\frac{(\frac{1}{4}x^{2})^{2k}}{k!}}}
\Kelvinber{\nu}@@{x}\Kelvinber{\nu}'@@{x}+\Kelvinbei{\nu}@@{x}\Kelvinbei{\nu}'@@{x} = \tfrac{1}{2}(\tfrac{1}{2}x)^{2\nu-1}\sum_{k=0}^{\infty}\frac{1}{\EulerGamma@{\nu+k+1}\EulerGamma@{\nu+2k}}\frac{(\frac{1}{4}x^{2})^{2k}}{k!}		 ℜ ⁡ ( ν + k + 1 ) > 0 , ℜ ⁡ ( ν + 2 ⁢ k ) > 0 formulae-sequence 𝜈 𝑘 1 0 𝜈 2 𝑘 0 {\displaystyle{\displaystyle\Re(\nu+k+1)>0,\Re(\nu+2k)>0}} 
KelvinBer(nu, x)*diff( KelvinBer(nu, x), x$(1) )+ KelvinBei(nu, x)*diff( KelvinBei(nu, x), x$(1) ) = (1)/(2)*((1)/(2)*x)^(2*nu - 1)* sum((1)/(GAMMA(nu + k + 1)*GAMMA(nu + 2*k))*(((1)/(4)*(x)^(2))^(2*k))/(factorial(k)), k = 0..infinity)

KelvinBer[\[Nu], x]*D[KelvinBer[\[Nu], x], {x, 1}]+ KelvinBei[\[Nu], x]*D[KelvinBei[\[Nu], x], {x, 1}] == Divide[1,2]*(Divide[1,2]*x)^(2*\[Nu]- 1)* Sum[Divide[1,Gamma[\[Nu]+ k + 1]*Gamma[\[Nu]+ 2*k]]*Divide[(Divide[1,4]*(x)^(2))^(2*k),(k)!], {k, 0, Infinity}, GenerateConditions->None]
Failure Successful
Failed [25 / 30]
Result: .71978298e-2-.3037583875e-1*I
Test Values: {nu = 1/2*3^(1/2)+1/2*I, x = 3/2}


Result: .607273780e-1-.1071579728*I
Test Values: {nu = 1/2*3^(1/2)+1/2*I, x = 2}

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Failed [3 / 30]
Result: Indeterminate
Test Values: {Rule[x, 1.5], Rule[ν, -2]}


Result: Indeterminate
Test Values: {Rule[x, 0.5], Rule[ν, -2]}

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10.65.E9 ( ber ν ′ ⁡ x ) 2 + ( bei ν ′ ⁡ x ) 2 = ( 1 2 ⁢ x ) 2 ⁢ ν - 2 ⁢ ∑ k = 0 ∞ 2 ⁢ k 2 + 2 ⁢ ν ⁢ k + 1 4 ⁢ ν 2 Γ ⁡ ( ν + k + 1 ) ⁢ Γ ⁡ ( ν + 2 ⁢ k + 1 ) ⁢ ( 1 4 ⁢ x 2 ) 2 ⁢ k k ! superscript diffop Kelvin-ber 𝜈 1 𝑥 2 superscript diffop Kelvin-bei 𝜈 1 𝑥 2 superscript 1 2 𝑥 2 𝜈 2 superscript subscript 𝑘 0 2 superscript 𝑘 2 2 𝜈 𝑘 1 4 superscript 𝜈 2 Euler-Gamma 𝜈 𝑘 1 Euler-Gamma 𝜈 2 𝑘 1 superscript 1 4 superscript 𝑥 2 2 𝑘 𝑘 {\displaystyle{\displaystyle\left(\operatorname{ber}_{\nu}'x\right)^{2}+\left(% \operatorname{bei}_{\nu}'x\right)^{2}=(\tfrac{1}{2}x)^{2\nu-2}\sum_{k=0}^{% \infty}\frac{2k^{2}+2\nu k+\frac{1}{4}\nu^{2}}{\Gamma\left(\nu+k+1\right)% \Gamma\left(\nu+2k+1\right)}\frac{(\frac{1}{4}x^{2})^{2k}}{k!}}}
\left(\Kelvinber{\nu}'@@{x}\right)^{2}+\left(\Kelvinbei{\nu}'@@{x}\right)^{2} = (\tfrac{1}{2}x)^{2\nu-2}\sum_{k=0}^{\infty}\frac{2k^{2}+2\nu k+\frac{1}{4}\nu^{2}}{\EulerGamma@{\nu+k+1}\EulerGamma@{\nu+2k+1}}\frac{(\frac{1}{4}x^{2})^{2k}}{k!}		 ℜ ⁡ ( ν + k + 1 ) > 0 , ℜ ⁡ ( ν + 2 ⁢ k + 1 ) > 0 formulae-sequence 𝜈 𝑘 1 0 𝜈 2 𝑘 1 0 {\displaystyle{\displaystyle\Re(\nu+k+1)>0,\Re(\nu+2k+1)>0}} 
(diff( KelvinBer(nu, x), x$(1) ))^(2)+(diff( KelvinBei(nu, x), x$(1) ))^(2) = ((1)/(2)*x)^(2*nu - 2)* sum((2*(k)^(2)+ 2*nu*k +(1)/(4)*(nu)^(2))/(GAMMA(nu + k + 1)*GAMMA(nu + 2*k + 1))*(((1)/(4)*(x)^(2))^(2*k))/(factorial(k)), k = 0..infinity)

(D[KelvinBer[\[Nu], x], {x, 1}])^(2)+(D[KelvinBei[\[Nu], x], {x, 1}])^(2) == (Divide[1,2]*x)^(2*\[Nu]- 2)* Sum[Divide[2*(k)^(2)+ 2*\[Nu]*k +Divide[1,4]*\[Nu]^(2),Gamma[\[Nu]+ k + 1]*Gamma[\[Nu]+ 2*k + 1]]*Divide[(Divide[1,4]*(x)^(2))^(2*k),(k)!], {k, 0, Infinity}, GenerateConditions->None]
Failure Successful
Failed [3 / 30]
Result: Float(undefined)+Float(undefined)*I
Test Values: {nu = -2, x = 3/2}


Result: Float(undefined)+Float(undefined)*I
Test Values: {nu = -2, x = 1/2}

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Failed [3 / 30]
Result: Indeterminate
Test Values: {Rule[x, 1.5], Rule[ν, -2]}


Result: Indeterminate
Test Values: {Rule[x, 0.5], Rule[ν, -2]}

... skip entries to safe data
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